Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 356 / 1

Let $w_{ij}$ be the transition rate from $i$ to $j$ and define the probability current $J_{ij}=w_{ij}{P}_i-w_{ji}{P}_j=-J_{ji}$. The master equation is ${\dot{P}}_i=-{\sum }_j{J}_{ij}$. Differentiating the Shannon entropy $S_{\mathrm{sys}}=-{\sum }_i{P}_i\log P_i$ and symmetrizing givesAdding the entropy flow to the environment produces the nonnegative entropy production rate of a Markov chainThe paper's displayed $\sum P\log P$ is the negative of the thermodynamic system entropy, so its derivative has the opposite sign before the environmental contribution is added.

For the two-state chain,with $P_1+P_2=1$.

Eliminating $P_2=1-P_1$ gives ${\dot{P}}_1=\beta -(\alpha +\beta )P_1$. ThereforeWith $r=\alpha p-\beta (1-p)$,

The only independent current isHence the total transient entropy production isIts relaxational part may be writtenwhile the steady or housekeeping entropy production is zero. Both the current and total production vanish as $t\to \infty$.

At stationarity,Thus every edge current vanishes and the two-state system satisfies detailed balance. A network containing only one undirected edge cannot sustain a stationary cycle current.

With indices understood cyclically,The circulant generator has eigenvalues $0$ and $-3\varphi \pm i\sqrt{3}\psi$, where $\varphi =(\alpha +\beta )/2$ and $\psi =(\alpha -\beta )/2$. Expanding the initial vector $(1,0,0)$ in its three Fourier eigenvectors gives

For the oriented edge $i\to j=i+1(\mathrm{mod}3)$, $J_i=\alpha P_i-\beta P_j$. Substitution in the Markov-chain entropy production givesSince ${\sum }_i{J}_i=\alpha -\beta$,The second term is relaxational and vanishes for the uniform stationary distribution. The steady entropy exported to the environment is therefore

At stationarity $P_i=1/3$, but the cycle current is $J_i=(\alpha -\beta )/3$. Therefore the chain is out of detailed balance exactly whenThe nonzero cycle affinity $3\log (\alpha /\beta )$ sustains the stationary current and positive housekeeping entropy production.